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 A178640 Triangle T(n,k) with the coefficient [x^k] of the series (-1)^(n+1) *(x-1)^(n+1)* sum_{j>=0} (5+8*j)^n*x^j in row n, column k. 1

%I

%S 1,5,3,25,94,9,125,1697,1223,27,625,25436,57926,14236,81,3125,352543,

%T 1903218,1513438,159593,243,15625,4717434,52306583,95276588,34660263,

%U 1766458,729,78125,62123517,1301287905,4593751457,3854897607,738035607

%N Triangle T(n,k) with the coefficient [x^k] of the series (-1)^(n+1) *(x-1)^(n+1)* sum_{j>=0} (5+8*j)^n*x^j in row n, column k.

%C Row sums are 1, 8, 128, 3072, 98304, 3932160, 188743680, 10569646080, 676457349120,

%C 48704929136640, 3896394330931200,....

%C The two main parameters (5,8) in the definition could be substituted by other pairs of consecutive Fibonacci numbers. Using the pair (0,1) or (1,1) yields A008292. Using the pair (1,2) yields A060187.

%F T(n,0) = 5^n = A000351(n).

%e 1;

%e 5, 3;

%e 25, 94, 9;

%e 125, 1697, 1223, 27;

%e 625, 25436, 57926, 14236, 81;

%e 3125, 352543, 1903218, 1513438, 159593, 243;

%e 15625, 4717434, 52306583, 95276588, 34660263, 1766458, 729;

%e 78125, 62123517, 1301287905, 4593751457, 3854897607, 738035607, 19469675, 2187;

%e 390625, 812215096, 30495345372, 189174172168, 303412512454, 137293837704, 15054569308, 214299832, 6561;

%p A178640 := proc(n,k) (-1)^(n+1)*(x-1)^(n+1)*add( (5+8*j)^n*x^j,j=0..k) ; coeftayl(%,x=0,k) ; end proc: # R. J. Mathar, Apr 05 2011

%t Clear[m, m0, t, n, k]

%t m0 = {{1, 1}, {1, 0}}

%t m[l_] := MatrixPower[m0, l]

%t t[l_, k_] = If[l == 0, 1, m[l][[1, 1]]*k + m[l][[1, 2]]]

%t p[x_, n_, l_] := (-1)^(n + 1)*(-1 + x)^(n + 1)*Sum[t[l, k]^ n*x^k, {k, 0, Infinity}]

%t Table[Flatten[Table[CoefficientList[FullSimplify[ExpandAll[p[x, n, l]]], x], {n, 0, 10}]], {l, 0, 10}]

%K nonn,tabl

%O 0,2

%A _Roger L. Bagula_, May 31 2010

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